There are many flows of practical importance where both Tollmien—Schlichting waves and Taylor—Gortler vortices are possible causes of transition to turbulence. In this paper, the effect of fully nonlinear Taylor—Gortler vortices on the growth of small-amplitude Tollmien—Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external-boundary-layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien—Schlichting waves considered have the asymptotic structure of lower-branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the former modes are simpler to analyse and more relevant to the boundary-layer problem. The effect of fully nonlinear Taylor—Görtler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.